octonions and the rolling ball
John Baez
Subriemannian Geometries, Their Geodeisics and Applications"
and I came across an interesting puzzle.
I don't have the energy to explain what this means -
(see pages 87-88 in his book if you're curious - but
in an 1910 paper Elie Cartan worked out an invariant
of "(2,3,5) distributions" on a manifold, and when this
invariant vanishes he showed the exceptional Lie group
G2 acts on the manifold in question.
Montgomery shows that an example of this situation
arises when we consider the phase space of a ball
rolling on a table. This manifold is 5-dimensional:
it takes 2 numbers to say where the center of the ball is,
and 3 more to describe the rotational degrees of freedom
of the ball. There is a 3-dimensional sub-bundle of
the tangent bundle which describes the ways the ball
can roll without slipping, and a 2-dimensional sub-bundle
which describes the ways the ball can roll without
slipping or "spinning", i.e. rotating about the vertical
axis. Perhaps this enough for you to guess why this
gives a "(2,3,5) distribution" in Cartan's terminology.
So, Montgomery poses the following open question:
Find an explicit geometric or physical description of the
G2 action on the ball-table system.
This is interesting, but it's especially interesting
to me because G2 is the automorphism group of the octonions!
Answering Montgomery's question would for the first time
give an example of the octonions appearing in real-world
physics (as opposed to unproven theories of particle physics,
like string theory).
But it's also puzzling, because the lowest-dimensional
example I know of a manifold on which G2 acts nontrivially
is 6-dimensional: the unit sphere in the imaginary octonions.
Of course, if you can answer Montgomery's question I'd
be delighted. But I'd settle for any example of a 5-dimensional
manifold on which G2 acts in a nontrivial way.
Danny Ross Lunsford
> So, Montgomery poses the following open question:
>
> Find an explicit geometric or physical description of the
> G2 action on the ball-table system.
>
> This is interesting, but it's especially interesting
> to me because G2 is the automorphism group of the octonions!
> Answering Montgomery's question would for the first time
> give an example of the octonions appearing in real-world
> physics (as opposed to unproven theories of particle physics,
> like string theory).
>
> But it's also puzzling, because the lowest-dimensional
> example I know of a manifold on which G2 acts nontrivially
> is 6-dimensional: the unit sphere in the imaginary octonions.
That is fascinating, because it may be a physical example of
non-associativity - namely the non-holonomic character of the constraint
(ball rolls without slipping).
I'm betting the geometry you want is Plucker geometry (line coordinates).
This has a well-known relation to RP5. I'll investigate in more detail and
report.
-drl
James Dolan
<ba...@galaxy.ucr.edu> wrote:
|But it's also puzzling, because the lowest-dimensional example I know
|of a manifold on which G2 acts nontrivially is 6-dimensional: the unit
|sphere in the imaginary octonions.
well, here's a reference to a usenet post by some guy talking about a
5-dimensional manifold on which g2 acts non-trivially:
--
[e-mail address jdo...@math.ucr.edu]
Aaron Bergman
In article <ao531u$1ae$1...@glue.ucr.edu>, James Dolan wrote:
>
> http://groups.google.com/groups?q=dolan+%22dynkin%22+group:sci.physics.research+author:john+author:baez&hl=en&lr=&ie=UTF-8&oe=UTF-8&safe=off&scoring=d&selm=aapt9e%24lk1%241%40glue.ucr.edu&rnum=3
This is a bit off-topic, but it's useful. A google groups URL only
needs the selm= part. For example, the above can be shortened to:
<http://groups.google.com/groups?selm=aapt9e%24lk1%241%40glue.ucr.edu>
Aaron
Mark Horn
11-OCT-2002
I must admit that I am not sure what it means for G2 to act
non-trivially on a 5-D manifold, but is there a connection with the
fact that a spherical tiling with (p,q,r) symmetry group (a triangle
group with angles pi/p, pi/q, pi/r),
(p,q,r) = (2, 3, 5),
yields a dodecahedron (when our basepoint is pi/q), with each face of
the geometric object being 5-dimensional?
mark jonathan horn
ba...@galaxy.ucr.edu
Noam D. Elkies <elkie...@h.harvard.edu> wrote:
>For the compact real form of G2, I can believe that SU(3) is
>the proper subgroup of maximal dimension. For the split form,
>or for G2(C), the group has maximal parabolics P of dimension 9,
>so codimension 14-9=5. Fulton and Harris discuss these maximal
>parabolics around page 391 of their book _Representation Theory:
>A First Course_. For one of them, G/P is a quadric surface in P^6,
>which is presumably the octonions o such that o^2=0 (that is,
>Tr(o)=Norm(o)=0), modulo scaling.
Just to clarify: as you mentioned in your email to me, these
"octonions" are really the split octonions if we're working
with the split form of G2, or the complexified octonions
(aka bioctonions) if we're working with the complex form G2(C).
>This seems like a reasonable guess for John Baez's 5-manifold.
>Fulton-Harris indicates that the other G/P is harder to get
>one's hands on.
Naively I'd guess that for the split octonions, the
the equation Norm(o) = 0 amounts to something like
a^2 + b^2 + c^2 + d^2 - e^2 - f^2 - g^2 - h^2 = 0
where (a,b,c,d,e,f,g,h) are an 8-tuple of real numbers.
The condition Tr(o) = 0 says the "real part" of our split
octonion vanishes, and naively I'd guess this gives us
an equation like
b^2 + c^2 + d^2 - e^2 - f^2 - g^2 - h^2 = 0
in 7 variables. If we projectivize this we get a
manifold diffeomorphic to (S^2 x S^3) / Z_2 . It
seems a bit odd that this is the same projective
quadric we get from G/P where G = SO(4,4) and P is
a maximal parabolic, so maybe I'm screwing up, or
maybe this is one of those coincidences like how
S^7 is a homogenous space of both SO(8) and the
compact real form of G2.
Anyway, this space (S^2 x S^3) / Z_2 is a bit different,
but not *drastically* different, from the phase space
of a ball rolling on the plane - namely R^2 x (S^3 / Z_2).
So, maybe we're pretty close!
I'll work through it more carefully sometime and
straighten it out. Thanks a million!